(a) Estimate by a dimensional argument or otherwise the order of magnitude of the gravitational self-energy of the Sun, with M = 2 1033g

(a) Estimate by a dimensional argument or otherwise the order of magnitude of the gravitational self-energy of the Sun, with Mθ = 2 × 1033g and Rθ = 7 × 1010 cm. The gravitational constant G is 6.6 × 10-8 dyne cm2 g-2. The sell-energy will be negative referred to atoms at rest at infinite separation.
(b) Assume that the total thermal kinetic energy of the atoms in the Sun is equal to –¼ times the gravitational energy. This is the result of the virial theorem of mechanics. Estimate the average temperature of the Sun. Take the number of particles as I x 10. This estimate gives somewhat too low a temperature, because the density of the Sun is far from uniform. “The range in central temperature for different stars, excluding only those composed of degenerate matter for which the law of perfect gases does not hold (white dwarfs) and those which have excessively small average densities (giants and super giants), is between 1.5 and 3.0 × 107 degrees.”